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Probability Distributions for Analog-To-Target Distances

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  • 1 a Laboratoire des Sciences du Climat et de l’Environnement, UMR 8212 CNRS-CEA-UVSQ, Institut Pierre-Simon Laplace and Université Paris-Saclay, Gif-sur-Yvette, France
  • | 2 b Lab-STICC, UMR CNRS 6285, IMT Atlantique, Plouzané, France
  • | 3 c France Énergies Marines, Plouzané, France
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Abstract

Some properties of chaotic dynamical systems can be probed through features of recurrences, also called analogs. In practice, analogs are nearest neighbors of the state of a system, taken from a large database called the catalog. Analogs have been used in many atmospheric applications including forecasts, downscaling, predictability estimation, and attribution of extreme events. The distances of the analogs to the target state usually condition the performances of analog applications. These distances can be viewed as random variables, and their probability distributions can be related to the catalog size and properties of the system at stake. A few studies have focused on the first moments of return-time statistics for the closest analog, fixing an objective of maximum distance from this analog to the target state. However, for practical use and to reduce estimation variance, applications usually require not just one but many analogs. In this paper, we evaluate from a theoretical standpoint and with numerical experiments the probability distributions of the K shortest analog-to-target distances. We show that dimensionality plays a role on the size of the catalog needed to find good analogs and also on the relative means and variances of the K closest analogs. Our results are based on recently developed tools from dynamical systems theory. These findings are illustrated with numerical simulations of well-known chaotic dynamical systems and on 10-m wind reanalysis data in northwest France. Practical applications of our derivations are shown for forecasts of an idealized chaotic dynamical system and for objective-based dimension reduction using the 10-m wind reanalysis data.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Paul Platzer, paul.platzer@ifremer.fr

Abstract

Some properties of chaotic dynamical systems can be probed through features of recurrences, also called analogs. In practice, analogs are nearest neighbors of the state of a system, taken from a large database called the catalog. Analogs have been used in many atmospheric applications including forecasts, downscaling, predictability estimation, and attribution of extreme events. The distances of the analogs to the target state usually condition the performances of analog applications. These distances can be viewed as random variables, and their probability distributions can be related to the catalog size and properties of the system at stake. A few studies have focused on the first moments of return-time statistics for the closest analog, fixing an objective of maximum distance from this analog to the target state. However, for practical use and to reduce estimation variance, applications usually require not just one but many analogs. In this paper, we evaluate from a theoretical standpoint and with numerical experiments the probability distributions of the K shortest analog-to-target distances. We show that dimensionality plays a role on the size of the catalog needed to find good analogs and also on the relative means and variances of the K closest analogs. Our results are based on recently developed tools from dynamical systems theory. These findings are illustrated with numerical simulations of well-known chaotic dynamical systems and on 10-m wind reanalysis data in northwest France. Practical applications of our derivations are shown for forecasts of an idealized chaotic dynamical system and for objective-based dimension reduction using the 10-m wind reanalysis data.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Paul Platzer, paul.platzer@ifremer.fr
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